Mod-3-Cohomology of SymplecticGroup(6,4), a group of order 4106059776000

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • SymplecticGroup(6,4) is a group of order 4106059776000.
  • The group order factors as 218 · 34 · 53 · 7 · 13 · 17.
  • The group is defined by Group([(3,4,6)(5,8,13)(7,11,19)(9,15,26)(10,17,30)(12,21,37)(14,24,43)(16,28,50)(18,32,57)(20,35,63)(22,39,70)(23,41,74)(25,45,81)(27,48,87)(29,52,94)(31,55,100)(33,59,107)(34,61,111)(36,65,115)(38,68,123)(40,72,130)(42,76,137)(44,79,134)(46,83,149)(47,85,153)(49,89,160)(51,92,166)(53,96,172)(54,98,175)(56,102,181)(58,105,187)(60,109,194)(62,113,201)(64,116,206)(66,119,212)(67,121,215)(69,125,222)(71,128,228)(73,132,235)(75,135,240)(77,139,246)(78,141,249)(80,144,255)(82,147,260)(84,151,266)(86,155,273)(88,158,278)(90,162,284)(91,164,288)(93,167,281)(95,170,277)(97,174,283)(99,176,299)(101,179,305)(103,183,312)(104,185,267)(106,189,322)(108,192,328)(110,196,335)(112,199,341)(114,203,348)(117,208,356)(118,210,318)(120,213,157)(122,217,366)(124,220,372)(126,224,377)(127,226,380)(129,230,386)(131,233,390)(133,237,396)(136,242,404)(138,245,410)(140,247,413)(142,251,419)(143,253,423)(145,257,429)(146,259,433)(148,262,418)(150,264,412)(152,268,441)(154,271,302)(156,275,451)(159,279,448)(161,282,446)(163,286,171)(165,274,449)(168,291,468)(169,272,214)(173,296,475)(177,301,481)(178,303,483)(180,307,489)(182,310,495)(184,314,502)(186,317,506)(188,320,306)(190,324,517)(191,326,521)(193,330,528)(195,333,533)(197,337,331)(198,339,541)(200,343,548)(202,346,554)(204,350,559)(205,300,373)(207,354,566)(209,358,573)(211,361,225)(216,364,582)(218,368,587)(219,370,315)(221,374,595)(223,376,598)(227,382,605)(229,384,607)(231,387,611)(232,389,614)(234,336,536)(236,394,498)(238,397,392)(239,399,628)(241,402,633)(243,406,640)(244,408,644)(248,287,463)(250,417,657)(252,421,661)(256,427,670)(258,431,630)(261,436,682)(263,297,476)(265,439,687)(269,443,693)(270,445,482)(276,293,471)(280,454,705)(285,460,713)(289,464,294)(290,466,722)(292,469,726)(295,473,725)(298,478,738)(304,485,334)(308,491,757)(309,493,760)(311,497,764)(313,500,721)(316,351,561)(319,509,747)(321,512,371)(323,515,505)(325,519,791)(327,523,794)(329,526,799)(332,531,805)(338,539,549)(340,543,821)(342,546,766)(344,550,829)(345,552,831)(347,555,834)(349,557,734)(352,563,840)(353,565,843)(355,568,786)(357,571,851)(359,575,857)(360,577,378)(362,579,379)(363,581,530)(365,584,749)(367,585,867)(369,589,873)(375,596,883)(381,603,889)(383,606,385)(388,612,825)(391,617,893)(393,532,807)(395,622,900)(398,626,773)(400,572,853)(401,631,917)(403,635,921)(405,638,924)(407,642,928)(409,632,918)(411,648,678)(414,651,939)(415,653,940)(416,655,756)(420,660,949)(422,662,951)(424,665,916)(426,668,574)(428,672,963)(430,674,854)(432,676,664)(434,679,973)(435,680,974)(437,492,758)(438,685,699)(440,689,982)(442,691,984)(444,695,902)(447,698,992)(450,701,456)(452,459,711)(453,704,999)(455,707,991)(457,697,990)(458,646,933)(461,714,1005)(462,716,1007)(465,720,1014)(467,724,1017)(470,728,556)(472,719,1012)(474,733,527)(477,737,1030)(479,740,891)(480,742,597)(484,745,663)(486,748,604)(487,750,652)(488,752,1043)(490,755,923)(494,761,1049)(496,623,908)(499,767,1009)(501,770,765)(503,772,1060)(504,715,804)(507,776,989)(508,777,599)(510,551,830)(511,780,562)(513,782,1066)(514,784,591)(516,787,1071)(518,790,1074)(520,792,1029)(522,608,800)(524,796,1080)(525,798,833)(529,802,1084)(534,809,812)(535,811,1078)(537,814,1026)(538,816,1096)(540,818,815)(542,690,835)(544,823,1101)(545,643,762)(547,826,717)(553,832,1107)(558,836,696)(560,838,896)(564,842,1115)(567,844,774)(569,847,1119)(570,849,1121)(576,858,1129)(578,859,600)(580,862,860)(583,865,1136)(586,869,785)(588,753,976)(590,875,1145)(592,878,1148)(593,880,1151)(594,881,884)(601,887,886)(602,727,1021)(609,894,1085)(610,694,987)(613,898,1158)(615,901,1163)(616,903,688)(618,837,1111)(619,739,937)(620,906,983)(621,907,1170)(624,909,1172)(625,864,1103)(627,912,819)(629,914,1130)(634,879,1149)(636,922,972)(637,759,964)(639,925,993)(641,927,1186)(645,931,1142)(647,788,1072)(649,936,945)(650,938,1194)(654,941,1197)(656,944,1200)(658,947,932)(659,783,1068)(666,955,1211)(669,959,1215)(671,962,1219)(673,965,1126)(675,968,1222)(677,970,1181)(681,975,919)(683,709,1001)(684,978,960)(686,979,895)(692,732,1025)(700,994,1153)(702,997,1213)(703,712,935)(706,1000,1191)(708,877,1138)(710,1003,1236)(718,1010,1187)(723,730,1023)(729,763,1052)(731,888,1154)(735,1027,1246)(736,1028,1065)(741,899,1160)(743,1035,841)(744,1036,1253)(746,905,797)(751,820,1098)(754,1045,1168)(768,942,986)(769,806,1086)(771,1058,1051)(775,1063,817)(778,1064,885)(779,828,1038)(781,1046,1259)(789,1073,1134)(793,1077,1270)(795,1079,1196)(801,1016,1167)(803,1056,1108)(808,1089,1277)(810,1090,1205)(813,890,1156)(822,897,1040)(824,934,910)(827,1105,1091)(839,1112,1276)(845,1117,1180)(846,1118,920)(848,1120,1293)(852,1124,1127)(855,1128,1298)(856,1125,1297)(861,1132,1131)(863,1133,1041)(866,1042,1258)(868,1139,1069)(870,1141,1302)(871,1044,1227)(872,1143,1303)(874,1144,1279)(876,1147,1305)(882,1113,1034)(892,1013,1019)(904,1166,1195)(911,1175,1097)(913,1177,1312)(915,1179,1314)(926,1185,1244)(929,1189,1162)(930,1190,946)(943,1199,1067)(948,1203,1217)(950,1146,1150)(952,1206,1241)(953,1208,954)(956,1209,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2,1294,1348,1358,1338,970,1225,701,996,607,892)(222,375,597,842,1116,1215,1118,1292,1320,242,405,639,926,555,387)(224,378,600,886,1153,716,1008,1090,500,769,1056,533,808,941,1198)(233,391,618,356,570,850,1122,1295,1349,1291,1323,1335,938,1195,1319)(246,412,650,521,384,608,893,979,1133,791,1076,473,732,1026,1112)(247,414,652,867,1138)(249,416,656,945,1202,631,436,683,977,1222,1325,1355,1289,1313,918)(264,438,686,980,912)(266,440,690,784,1069)(278,449,700,995,548,364,583,866,427,671,660,928,1188,1317,374)(284,459,712,301,482)(288,451,702,998,780,399,629,915,1180,925,1021,606,396,624,376)(307,490,756,1047,1014,799,1082,1273,1107,1084,1052,1262,1334,386,610)(317,507,563,841,1114,1179,1199,1293,755,419,659,948,1204,1049,1080)(333,534,810,1091,1170)(358,574,856,931,1191)(361,579,861,705,933,1192,875,1146,554,833,1109,491,737,1031,1246)(366,512,781,973,982)(390,616,904,1167,901)(402,634,469,727,1022,1245,497,765,1054,1264,1340,523,795,821,1100)(431,675,969,1224,1327)(443,694,988,776,742,1034,1251,1208,1321,994,726,1020,1244,1329,1310)(464,719,1013,1111,757,1028,1247,1233,1239,1096,1280,1345,1357,1362,1365)(468,725,1019,1060,823,1102,1283,1331,1025,909,1173,1311,1352,1359,1363)(475,735,577,859,1131)(476,736,1029,1186,493,603,890,1101,1236,1302,1027,777,1064,1132,990)(531,806,1087,1274,1277)(584,633,920,1182,898,1159,676,955,1212,999,1203,834,1110,796,1058)(611,896,830,655,943,1120,1278,1121,665,954,1210,698,993,761,1050)(614,900,1162,1308,1343)(617,905,1168,782,1067,738,1032,1066,1143,1304,1336,1148,978,1194,1086)(648,935,1193,1318,1354,1360,1364,711,1004,1098,1282,857,1129,1296,1324)(687,981,1230,805,1085,772,740,1033,1250,1337,1356,1361,1346,1351,1341)(689,983,1158,1307,1350,963,792,1017,1241,1275,1256,1339,1220,1314,1144)(707,869,1140)(720,826,1104,1284,1287,940,1196,1309,883,840,1113,1288,1211,1226,722)(814,1094,851,1123,1219,1145,1232,1316,1156,1136,1300,1124,959,1216,939)(914,1178,974,1001,1160)]).
  • It is non-abelian.
  • It has 3-Rank 3.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 3, respectively.

Structure of the cohomology ring

The computation was based on 3 stability conditions for H*(Normalizer(SymplecticGroup(6,4),Centre(SylowSubgroup(SymplecticGroup(6,4),3))); GF(3)).

General information

  • The cohomology ring is of dimension 3 and depth 3.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1)·((1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6) · (1  −  t  +  t2  −  t3  +  t4  −  t5  +  t6  −  t7  +  t8  −  t9  +  t10))
    ( − 1  +  t)3 · (1  +  t  +  t2) · (1  +  t2)3 · (1  −  t2  +  t4) · (1  +  t4)
  • The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 6 minimal generators of maximal degree 12:

  1. a_3_0, a nilpotent element of degree 3
  2. b_4_0, an element of degree 4
  3. a_7_0, a nilpotent element of degree 7
  4. b_8_0, an element of degree 8
  5. a_11_0, a nilpotent element of degree 11
  6. c_12_0, a Duflot element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 3 "obvious" relations:
   a_3_02, a_7_02, a_11_02

Apart from that, there are no relations.

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 21 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 0 and the last generator in degree 12.
  • The following is a filter regular homogeneous system of parameters:
    1.  − b_4_0·b_8_04 − b_4_03·b_8_03 − b_4_09 + b_8_03·c_12_0 + b_4_02·b_8_02·c_12_0
         + c_12_03, an element of degree 36
    2. b_8_06 − b_4_02·b_8_05 − b_4_04·b_8_04 − b_4_06·b_8_03
         − b_4_03·b_8_03·c_12_0 + b_4_05·b_8_02·c_12_0 + b_8_03·c_12_02
         − b_4_02·b_8_02·c_12_02 + b_4_03·c_12_03, an element of degree 48
    3. b_8_0, an element of degree 8
  • A Duflot regular sequence is given by c_12_0.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 89].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. b_4_03 + c_12_0, an element of degree 12
    2. b_4_0, an element of degree 4
    3. b_8_0, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(Normalizer(SymplecticGroup(6,4),Centre(SylowSubgroup(SymplecticGroup(6,4),3))); GF(3))

  1. a_3_0 → a_3_0
  2. b_4_0 → b_4_0
  3. a_7_0 → a_7_5 + a_7_1 + b_4_1·a_3_0 + b_4_0·a_3_1
  4. b_8_0 → b_8_3 − b_4_0·b_4_1
  5. a_11_0 → a_11_8 − b_4_1·a_7_1 + b_4_12·a_3_0 − b_4_0·b_4_1·a_3_1
  6. c_12_0 → b_4_1·b_8_3 + b_4_13 + b_4_0·b_4_12 + c_12_6

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_3_0 → 0, an element of degree 3
  2. b_4_0 → 0, an element of degree 4
  3. a_7_0 → 0, an element of degree 7
  4. b_8_0 → 0, an element of degree 8
  5. a_11_0 → 0, an element of degree 11
  6. c_12_0 → c_2_06, an element of degree 12

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_0 →  − c_2_2·a_1_1, an element of degree 3
  2. b_4_0 →  − c_2_22, an element of degree 4
  3. a_7_0 → 0, an element of degree 7
  4. b_8_0 → 0, an element of degree 8
  5. a_11_0 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  6. c_12_0 → c_2_12·c_2_24 + c_2_14·c_2_22 + c_2_16, an element of degree 12

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

  1. a_3_0 →  − c_2_5·a_1_1 + c_2_5·a_1_0 − c_2_4·a_1_2 − c_2_4·a_1_1 + c_2_3·a_1_2, an element of degree 3
  2. b_4_0 → c_2_4·c_2_5 − c_2_42 − c_2_3·c_2_5, an element of degree 4
  3. a_7_0 →  − c_2_53·a_1_1 + c_2_53·a_1_0 − c_2_4·c_2_52·a_1_1 + c_2_4·c_2_52·a_1_0
       − c_2_42·c_2_5·a_1_2 − c_2_42·c_2_5·a_1_0 + c_2_3·c_2_52·a_1_1 − c_2_3·c_2_52·a_1_0
       − c_2_3·c_2_4·c_2_5·a_1_2 + c_2_3·c_2_4·c_2_5·a_1_1 − c_2_3·c_2_42·a_1_2
       − c_2_32·c_2_5·a_1_2 + c_2_33·a_1_2, an element of degree 7
  4. b_8_0 →  − c_2_4·c_2_53 + c_2_42·c_2_52 + c_2_3·c_2_53 + c_2_3·c_2_4·c_2_52
       − c_2_3·c_2_42·c_2_5 + c_2_32·c_2_52 + c_2_33·c_2_5, an element of degree 8
  5. a_11_0 →  − c_2_55·a_1_1 + c_2_55·a_1_0 + c_2_4·c_2_54·a_1_2 − c_2_4·c_2_54·a_1_1
       − c_2_4·c_2_54·a_1_0 + c_2_42·c_2_53·a_1_2 + c_2_42·c_2_53·a_1_0
       − c_2_3·c_2_54·a_1_2 − c_2_3·c_2_54·a_1_1 + c_2_3·c_2_54·a_1_0
       − c_2_3·c_2_4·c_2_53·a_1_2 − c_2_3·c_2_4·c_2_53·a_1_1 + c_2_3·c_2_42·c_2_52·a_1_0
       + c_2_3·c_2_43·c_2_5·a_1_0 + c_2_3·c_2_44·a_1_0 − c_2_32·c_2_53·a_1_2
       + c_2_32·c_2_4·c_2_52·a_1_1 + c_2_32·c_2_42·c_2_5·a_1_2 − c_2_32·c_2_43·a_1_2
       − c_2_32·c_2_43·a_1_1 + c_2_33·c_2_52·a_1_1 − c_2_33·c_2_52·a_1_0
       − c_2_33·c_2_4·c_2_5·a_1_2 + c_2_33·c_2_4·c_2_5·a_1_1 + c_2_33·c_2_4·c_2_5·a_1_0
       − c_2_33·c_2_42·a_1_2 − c_2_33·c_2_42·a_1_0 + c_2_34·c_2_5·a_1_2
       + c_2_34·c_2_5·a_1_1 − c_2_34·c_2_5·a_1_0 + c_2_34·c_2_4·a_1_2 + c_2_34·c_2_4·a_1_1
       + c_2_35·a_1_2, an element of degree 11
  6. c_12_0 → c_2_56 + c_2_4·c_2_55 − c_2_42·c_2_54 − c_2_3·c_2_55 + c_2_3·c_2_4·c_2_54
       − c_2_3·c_2_42·c_2_53 + c_2_32·c_2_54 + c_2_32·c_2_42·c_2_52
       + c_2_32·c_2_43·c_2_5 + c_2_32·c_2_44 + c_2_33·c_2_53 − c_2_33·c_2_4·c_2_52
       + c_2_33·c_2_42·c_2_5 + c_2_34·c_2_52 − c_2_34·c_2_4·c_2_5 + c_2_34·c_2_42
       − c_2_35·c_2_5 + c_2_36, an element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010